Heat the vegetable oil in a large pot or Dutch oven over medium-high heat until the oil shimmers. Add the fresh shiitakes and cook, stirring occasionally, until they are browned and tender, about 5 to 7 minutes.

Stir in the water, mirin, rice vinegar, soy sauce, garlic, ginger, dried shiitakes, sesame oil, and chili-garlic sauce. Bring the mixture to a simmer, then reduce the heat to medium-low and cook until the liquid has reduced by about a third and the flavors are concentrated, about 15 to 20 minutes.

While the sauce is reducing, bring a separate large pot of salted water to a boil. Add the mustard greens and cook until they just begin to wilt, about 1 minute.

Add the udon noodles to the pot with the greens and cook until the noodles are heated through and tender, about 2 to 3 minutes.

Drain the noodles and greens well. Remove and discard the smashed garlic and ginger from the sauce pot, then add the noodles and greens to the sauce, tossing until well coated. Serve hot with salt and pepper to taste.